Small multiplier after initial approximation for operations with increasing precision

ABSTRACT

In an aspect, a processor includes circuitry for iterative refinement approaches, e.g., Newton-Raphson, to evaluating functions, such as square root, reciprocal, and for division. The circuitry includes circuitry for producing an initial approximation; which can include a LookUp Table (LUT). LUT may produce an output that (with implementation-dependent processing) forms an initial approximation of a value, with a number of bits of precision. A limited-precision multiplier multiplies that initial approximation with another value; an output of the limited precision multiplier goes to a full precision multiplier circuit that performs remaining multiplications required for iteration(s) in the particular refinement process being implemented. For example, in division, the output being calculated is for a reciprocal of the divisor. The full-precision multiplier circuit requires a first number of clock cycles to complete, and both the small multiplier and the initial approximation circuitry complete within the first number of clock cycles.

BACKGROUND Field

In one aspect, the following relates to microprocessormicroarchitecture, and in an example particular aspect, toimplementations of divide and square root functions in hardware.

Related Art

An important aspect of computation is the ability to perform arithmetic.Processors, such as generally programmable processors, digital signalprocessors, graphics processors, generally have a capability to performarithmetic, such as one or more of integer, fixed, and floating pointarithmetic. The performance of such arithmetic can be controlled byarithmetic instructions, which can vary among different architectures,but for the sake of example, can include addition, subtraction,multiplication, division, and square root instructions. A particularimplementation of such instructions may involve decomposing suchoperations into operations that are supported on a particular hardwareimplementation. For example, a particular implementation may not haveseparate circuitry implementing a floating point multiplier (or moregenerally, a floating point math unit), and as such, a floating pointmultiply instruction may be implemented by emulating the instruction inmicrocode on the processor, within the operating system, or in compileduser-space code. Such emulation is always much slower than a dedicatedhardware floating point unit. However, a hardware floating point unitconsumes area on a semiconductor substrate, and hence increases cost.

Some hardware arithmetic units implement divide and square rootoperations by using an iterative refinement from an initialapproximation, such as an implementation of Newton-Raphson. For example,dividing term a by term b (i.e., a/b) can be performed by finding thereciprocal of term b (1/b) using Newton-Raphson, and then multiplyingthat reciprocal by term a. Implementations of Newton-Raphson ofteninvolve using a LookUp Table (LUT) indexed by a portion of term b toproduce an initial approximation of the reciprocal of b, which is thenrefined through an appropriate number of iterations of theNewton-Raphson procedure.

For example, FIG. 1 depicts an operand 15, from which some portion ofindex bits 20 (some number of the most significant bits of the mantissaof operand 15) are used to index a LUT 25. In response to index bits 20,k bits of output 28 are produced. An implementation of LUT 25 may usearound 9 bits of the mantissa of operand 15, and the LUT 25 in such casewould generally provide k=7 or k=8 output bits. In other words, theinitial approximation of the reciprocal includes about 7 to 8 bits ofprecision. The final result of the calculation, however, generally isexpected to be accurate to 24 bits of precision for single-precisionfloating point and 53 bits for double-precision floating point. The kbits of output 28 are expanded in an expander 32 to a full size of afull-precision multiplier circuit 35 by padding zeroes to a leastsignificant bit side. Then, that input can be multiplied with anotherselected input. In the context of Newton Raphson, for the firstiteration, that input is the divisor (Term b). A complement of theoutput of that multiplication is then multiplied with the expandedinitial approximation to produce a revised approximation for asubsequent iteration. A number of bits of precision approximatelydoubles with each iteration. However, the output size of the multiplieris the same, regardless of a number of bits of precision actuallyavailable in each iteration.

FIG. 2 depicts a method of using the circuit of FIG. 1 that is generalto any algorithm that uses a table lookup for an initial low-precisionestimate of a value used in a multiplication. At 80, an operand isreceived, and at 81, a portion of the operand (e.g., a part of amantissa) is used to index a LUT. At 82, the LUT outputs k bits ofinitial approximation. At 83, those bits are padded to a size of inputto a multiplier that supports full-precision inputs. At 84, the expandedinitial approximation is provided to the multiplier, and at 85 anotherfull-size term is provided to the multiplier. At 86, the multiplierperforms a full-precision multiplication on the provided terms. A numberof bits of precision resulting from the multiplication depends on thesize of k and the number of bits of precision in the other term. Atypical value of k is 7 bits.

Most high-speed designs for multipliers use multiple clocks to fullyperform the multiplication of two mantissas. FIG. 3 depicts that amultiplier requires 3 clocks to produce a result that can be fed backfor further iteration in a Newton-Raphson algorithm. The Carry SaveAdder (CSA) requires 2 clocks and the Carry Lookahead Adder (CLA)requires 1 clock. In such a case, if the initial approximation requiresone clock to produce, then a 2-clock delay is required to synchronizethe start of the first multiplication for the Newton Raphson method withsubsequent multiplications.

The circuitry to implement a multiplier is a large structure and a carrysave adder (CSA) dominates an overall area of a multiplier. Therefore,it has been conventional to implement one multiplier and use thatmultiplier for as many different operations as possible.

SUMMARY

Because of the table lookup involved in operations that use an initialapproximation (e.g., divide and square root), which typically takes atleast one clock, a scheduling problem arises when using a singlemultiplier unit for regular multiplications and for operations usingiterative refinement, such as divide and square root operations.

One aspect relates to an apparatus, such as a processor, for arithmeticcalculation. The apparatus includes a source for an initiationapproximation of one term (such as a reciprocal). Such source caninclude a LUT, for example. The LUT produces an output with k bits ofprecision, where k is less than a number of bits of precision in a finalresult. The apparatus includes a full-precision, large multiplier and areduced-precision, small multiplier. The reduced precision smallmultiplier receives the k bits outputted from the LUT as a first termand multiplies that term with a sufficient number of bits from a secondterm to result in p bits of precision in an output result, where p>k.The small multiplier has less latency than the large multiplier, andcollectively, a sum of the latency to produce the initial approximationand the latency of the small multiplier is no greater than the latencyof the large multiplier (in terms of clock units). Such apparatus alsoincludes logic to control usage of the small multiplier and the largemultiplier to perform iterative refinement. In one implementation, thesmall multiplier described above is used for the first multiplication ofthe first iteration of a reciprocal calculation. Thereafter, the large,full-precision multiplier is used. Thus, such an apparatus includes botha full-precision multiplier and a separate reduced-precision multiplier.

A further aspect includes that an apparatus may provide a hierarchy ofhardware elements that each perform the same operation but maintaindifferent numbers of bits of precision in their respective outputs. Thehierarchy of hardware elements is configured so that outputs oflower-precision elements provide inputs to higher precision elements.Elements within the hierarchy, or some portion thereof, also mayselectively recycle an output either to itself, or to the input ofanother element. In order to calculate a final output value for asequence of operations, progressively greater precision outputs areachieved by starting from low-precision inputs to the lowest precisionhardware element, and allowing outputs of increasingly higher precisionto propagate through the hierarchy, where propagation also includesmultiple iterations through some subset of the hardware elements. In aparticular implementation, the hierarchy has two levels, alower-precision and a full-precision level. Embodiments according to thedisclosure include a processor that includes an instruction set thatincludes divide and square root instructions, wherein the divide andsquare root instructions are performed using implementations accordingto the disclosure. Some processor implementations may have otherinstructions that are performed using techniques disclosed herein. Otherimplementations may be provided, for example, in semi-programmable, orfixed-function logic.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 depicts a block diagram of some aspects of a prior artdivide/square root part of a hardware arithmetic unit;

FIG. 2 depicts a prior art method of using the circuitry depicted inFIG. 1 ;

FIG. 3 depicts wait states inserting to synchronize a start of a firstmultiply where a first clock was used for a table lookup; and

FIG. 4 depicts an example apparatus in which a small multiplier is usedto perform an initial low-precision multiplication;

FIGS. 5A and 5B depict example implementations of a pipelined flowwithin an apparatus according to the example of FIG. 4 ;

FIG. 6 and FIG. 7 depict different examples of using the smallmultiplier of FIG. 4 ;

FIG. 8 depicts an example construction of a small multiplier accordingto the disclosure;

FIGS. 9A-9E depict diagrams showing inputs and outputs through thevarious combinatorial logic elements of FIG. 8 .

DETAILED DESCRIPTION

As introduced above, aspects of the disclosure include arithmetic unitsthat have a large, full-precision multiplier and also a small, limitedprecision multiplier. Such arithmetic units include control logic thatcauses a first multiplication, which involves low precision inputs to beconducted within the limited precision multiplier, and a result of thatmultiplication then used as though it were an output of the fullprecision multiplier. Because the small multiplier is faster than thefull-precision multiplier, a total number of clocks to obtain both theinitial approximation and perform the first multiplication can be thesame as a number of clocks required for a full-precision multiplicationin the full-precision multiplier. The general convention of describingtiming for pipelines in terms of “clocks” or “clock cycles” is followedhere. However, it would be understood that this disclosure uses theseterms to describe relative delays for different circuitry in an exampleimplementation, rather than an absolute sense applying across allimplementations. Also, in general, a given portion of combinatoriallogic, or a result of some processing may be ready before an expirationof a given clock cycle for traversing that logic, or producing thatresult, even though the result may be sampled later.

Aspects of the disclosure are presented using an example Newton Raphsonimplementation. However, the identified aspects will find application inother contexts.

FIG. 4 depicts a full-precision multiplier 25 that includes amultiplexer 38 that can select from a mantissa of operand a 33, from amantissa of operand b 31 and from another input described below.Multiplier 25 also includes a multiplexer 39 that can selectively outputeither a result of small multiplier 55 or from a stage of full-precisionmultiplier 25, itself. For example, multiplier 25 includes a multi-stagepipeline, that, for example, includes two stages 41, 42 allocated to aCarry Save Adder (CSA) and a stage 44 allocated to a Carry LookaheadAdder (CLA). Finish stage 47 represents a variety of operations that maybe performed on an output from CLA stage 44, such as rounding andnormalization. Apparatus 50 also includes a divide/square root unit 66.Divide/square root unit 66 can be implemented by circuitry configured toperform operations and coupled as described below.

While mux 38 and mux 39 were depicted as being part of multiplier 25,these multiplexers and select logic can be formed from circuitry thatmay not be physically co-located with circuitry implementing portions ofmultiplier 25, even though in general, they may be.

A LookUp Table (LUT) 47 is coupled to receive a portion of bits(typically, a portion of the Most Significant Bits (MSBs)) of a valuefor which a reciprocal is to be calculated through iteration. LUT 47outputs an initial approximation 51 that has a known number of bits, k,of accuracy. Initial approximation 51 is input to a small multiplier 55.Initial approximation 51 is to be multiplied with an input 53 that iseither operand b, in one implementation, or a modified operand b inanother implementation. Modified operand b is explained below.

The output of small multiplier 55 provides output that is an input tomultiplexer 39. A current approximation storage 45 is coupled to receivean output from CLA 44 and is coupled to an input of multiplexer 38.Multiplexer 39 can have a separate input from small multiplier 55 andfrom CLA 44; control logic generating a select signal for mux 39 cancontrol output from mux 39 accordingly. A state machine 49 tracks stateassociated with iterative refinement operations. These elements arecharacterized as being included in divide/square root unit 66, sincethese elements are used in performing divide and square rootcalculations in the present example. However, physical realizations ofthese elements can be intermingled with other circuitry and the blockdiagram of FIG. 4 does not imply that a distinctive unit for divide andsquare root calculations or other calculations using iterativerefinement must be supplied or be identifiable in an implementation.

FIG. 5A depicts a pipeline that performs Newton Raphson to produce areciprocal of a value b, which is an operation that can be performed inorder to execute an instruction requesting a division of dividend a by b(i.e., a/b). A portion of b (for example, either 8 or 9 bits MostSignificant Bits) is supplied as an index to a LUT. The LUT provides(152) an initial approximation (x_(o)) of 1/b, which is 8 or 9 bits, andhas 7 or 8 bits of precision. This action requires one unit of time.Herein, a unit of time is called a “clock” for simplicity, but a unit oftime is a relative measure with respect to time required to take otheractions in the pipeline, not an absolute measure.

During the first iteration, n=0. In order to perform a firstmultiplication (154) of a first iteration, x_(o) is supplied to alow-precision (small) multiplier circuit (e.g., small multiplier 55 ofFIG. 4 ) which multiplies x_(o) by b (t₀=b*x₀). A next (and last)multiplication for the first iteration is performed (166) in a higherprecision multiplier (such as full precision multiplier 25). Thismultiplication is x₁=x₀*r₀ where r₀=˜t₀≈2−t₀. ˜t₀ is the complement oft₀ and is an approximation of 2−t_(n). This complement can be producedfaster than actually performing that subtraction, and hence is used inthis example implementation.

Lookup 152 and multiplication 154 collectively require K clocks (in oneexample, K=3), with no more time than used by multiplication 162. Aresult 168 of multiplication 166 yields a better approximation, X_(n+1),which is fed back to the first multiplication 162 of an iteration. Forexample, x₁ is produced by multiplication 166 at the end of the firstiteration, so that x₁ is used to produce t₁=b*x₁ in the seconditeration. Each subsequent multiply 162 or multiply 166 and complementrequires K clocks, respectively.

After an appropriate number of iterations, result 168 of multiplication166 is used as a sufficiently-accurate approximation of the reciprocalof b (1/b). The reciprocal of b is then used in a multiplication with ato complete the division. For single precision floating point, 24 bitsof precision are required; for double precision floating point, 53 bitsof precision are required; for single-width integer, 32 bits ofprecision are required; and for double-width integer, 64 bits arerequired. In this example, 7 or 8 bits of precision are provided (152)by the initial approximation, and each iteration doubles that precision.So, with 7 bits of precision and single precision floating point,multiplication 154 will be performed once, multiplication 162 will beperformed twice, and multiplication 166 will be performed 3 times. Whilethese examples do not describe extended float formats, the disclosuresherein could be adapted to an implementation supporting extendedfloating point formats. Such adaptation could include one or more ofproviding more precision in an initial approximation, and performing anadditional iteration.

With respect to the example of FIG. 4 , CLA 44 is a last stage ofmultiply 166, and output from CLA 44 can be sent to currentapproximation storage 45, if present, or directly to multiplexer 39, orboth, in some implementations. Current approximation storage 45 can beupdated with each updated approximation, x_(n+1). Some implementationsmay not require or use storage for a current approximation. On a finaliteration, CLA 44 would provide output to the finish stage 47, insteadof to multiply 162.

The arrangement of LUT 47, small multiplier 55, and full-precisionmultiplier 25 is designed to produce a result accurate to withinrequired precision within a pre-determined number of iterations. Statemachine 49 can count a number of iterations that have been performed inorder to determine when sufficient precision has been achieved in theoutput and then can cause a current value from CLA 44 to be provided tofinishing stage 47.

Operation of finishing stages to produce a final result of themultiplication and also to produce the result of the multiplicationbetween 1/b and a are not depicted, as these aspects are not central tothe disclosure.

FIG. 5B depicts an alternate implementation in which an x₁ with 14 or 16bits of precision is available after only one multiplication, such thatonly one refinement step is required to produce a single-precision valueand two refinement steps are required to produce a double-precisionvalue. This implementation generally is in accordance with an approachdescribed in Masayuki ITO, et al., “Efficient Initial Approximation forMultiplicative Division and Square Root by a Multiplication with OperandModification” IEEE Transactions on Computers, Vol. 46, No. 4, April 1997(“Ito” herein), with described modifications.

FIG. 5B depicts that 6 or 7 MSBs of b can be used as a LUT index toprovide (102) a startup term, x′, x′≠x₀. A first multiplication (104)produces x₁={circumflex over (b)}*x′. x₁ has 14 or 16 bits of precision.The multiplications 162 and 166 of FIG. 5A are the same in FIG. 5B,except that multiplication 162 is first performed, as t₁=b*x₁, nott₀=b*x₀ as in FIG. 5A, and these changes propagate through tomultiplication 166, and as necessary, to a subsequent iteration. Notethat, in both FIGS. 5A and 5B, x₂ is sufficient for single precision,while x₃ is sufficient for double precision. A principal difference isthat multiplication 162 and 166 are performed only once for singleprecision, and twice each for double precision.

The examples of FIGS. 5A and 5B were in the context of a division.Another example application of the disclosure is for square root. Squareroot can be implemented in accordance with either FIG. 5A or FIG. 5B.The value being produced through iterative refinement, for square root,is 1/√b. The first multiplication involving the initial approximationthereof is conducted in small multiplier 55, with subsequentmultiplications performed in full precision multiplier 25. For squareroot, step 166 is changed to compute x_(n+1)≈x_(n)(3−(x_(n)t_(n)))/2 andthis requires two executions of the full precision multiplier of Kclocks each.

FIGS. 6 and 7 depict two example alternatives to implementing lookuptable 47 and small multiplier 55 in accordance with FIGS. 5A and 5B,respectively. FIG. 6 depicts an implementation where 8 Most SignificantBits (MSBs) of operand b are used to index LUT 47, which outputs initialapproximation x₀ with 7 bits of precision. Small multiplier 55 thenperforms the first multiplication of the first iteration. Thismultiplication is between the 7 bits outputted from LUT 47 and 14 MSBsfrom operand b).

FIG. 7 depicts an alternative where LUT 47 produces a value x′ that isdesigned to be multiplied with a modified version of operand b({circumflex over (b)}). The multiplication between x′ and {circumflexover (b)} results directly in x₁, without requiring multiplication 166,as is required to produce x₁ in the approach depicted in FIG. 5A. Theimplementation of FIG. 6 requires two additional iterations to obtain asingle precision result, and three iterations to achieve a doubleprecision result. The implementation of FIG. 7 requires one iterationfor single precision and two for double precision. The modified versionof b ({circumflex over (b)}) can be formed as disclosed in Ito. Thevarious entries in the LUT (each an x′), are formed to provide anappropriate output value after multiplication with {circumflex over(b)}.

FIG. 8 depicts an example implementing a small multiplier 55 whenimplementing an approach according to FIG. 7 . Initial approximationcircuitry 125 outputs x′ as explained above, and modifier circuitry 127provides {circumflex over (b)}. In FIG. 8 , numbers in circles representeighths of a unit of time required to traverse a particular portion ofcombinatorial logic, and numbers in squares represent a sum of thosetimes within a particular clock cycle. FIG. 8 thus presents a moregranular view of how one clock cycle is allocated among differentcombinatorial logic elements within each stage of multiplier 55. A fanout 130 distributes portions of each operand across a set of boothmultiplexers 132. 3:2 compressors 134 provide an 8-wide output to 5:3and 3:2 compressors 136, resulting in a 5-wide output to 5:3 compressors138. Output from 5:3 compressors 138 is captured by flop 140,representing an end to one clock cycle. Flop 140 outputs to 3:2compressors 142 that outputs to a CLA 144. Pick/save mux 146 can outputa value to outputs to flop 148 and is operable to select CLA 144 outputx₁, such as for starting a divide or square root operation, or othervalue, such as a value needed for some iteration by the full precisionmultiplier 25, as an input to mux 39 (see FIG. 4 ). Thus, multiplier 55requires a total of two clock cycles to produce an output that can bereturned to fan out 130.

FIGS. 9A-9E show a progression through combinatorial logic elements ofFIG. 8 , and how outputs from one element are mapped to inputs of afollowing element. As explained above, small multiplier 55 does notimplement a full precision (full width) multiplication, but rather atruncated multiplication. FIG. 9A depicts a booth encoding of inputs to3:2 compressors 134 and that the multiplication is truncated as shown attruncation point 175. Each vertical line identifies a set of bits thatwill be presented as inputs to a particular 3:2 compressor in 3:2compressors 134. The stair-step formation shows a bottom edge of some ofthe partial products that are not produced by multiplier 55, which wouldbe produced by a full-precision multiplier. Dashed oval 176 identifiesbits that are relocated to the dashed oval 177. Bit locations identifiedwith the letter “S” show locations of sign bits or the negation of signbits. FIG. 9B shows what bits are input to each 5:3 and 3:2 compressorin compressors 136. Bits marked with “X” bypass to the next FIG. 9 ,while in some cases, a 5:3 compressor may actually only input 4 bits, asexplained in the legend associated with FIGS. 9A-9E. FIG. 9C depicts howbits are mapped to inputs of 5:3 compressors 138. FIG. 9D depicts howbits are mapped to inputs of 3:2 compressors 142. FIG. 9E depicts inputsto CLA 144, and each numbered pair of bits (0-8) identifies bits thatare generate positions for the CLA.

FIGS. 9A-9E thus show that a large portion of the carry save adderstructure that would be present in a full-precision multiplier is notpresent in an implementation of small multiplier 55. The absence ofthese structures allows small multiplier 55 to operate more quickly thanfull-precision multiplier 25.

Because least significant bits of partial products were not calculated,some error is present in the absence of potential carry-ins from thosepartial products. In other words, even though the output has 25 bits,not all of those bits can be considered accurate in this example,because carry-in bits that could have affected some portion of thelow-order bits in the 25-bit result are absent. For example, if all ofthe partial products resulted in a carry, then the calculatedlower-precision result could be lower by around twelve from what wouldhave been calculated in a full-precision multiplier. Assuming that onaverage half of the partial products would result in a carry-in, theresult would be expected to be too low by around 6. It is possible toinject a value to reduce this loss of precision. For example, a value 6can be injected (binary 110) into the least precision bits of the last3:2 compressor 142 (bits 24-22). However, the example of FIGS. 8 and 9implement a multiplier that has extra precision for what is required toproduce a single precision value in one iteration and a double precisionvalue in two iterations. This is because an output of multiplier 55 has25 bits, in this example, and at least 16 of these bits are accurate forx₁. Thus, after one iteration, the result will be accurate to 32 bits,which presents considerable margin for a single precision result, and isaccurate to 64 bits with one more iteration, which more than satisfiesdouble precision requirements.

As such, FIGS. 8 and 9 present an example of implementing the smallmultiplier to have as much precision as possible while also meeting thetiming requirement that the small multiplier and the LUT operationcomplete within the same number of time increments as one multiply ofthe full-precision multiplier (the time increments are also consideredin the context of a number of time increments that fit within one clock,where intermediate outputs are flopped and timing thus needs to besatisfied for those flops). It would be possible to design a smallmultiplier that has less margin of extra precision, and save some amountof area and power consumption, but it is not expected that such changewould result in reduction in delay sufficient to alter an approach tomaking circuit timing. Yet, different implementations may presentdifferent opportunities, and there may be some implementations in whichthose savings are important, and other implementations in which thosesavings may allow a different relative timing setup. Those of ordinaryskill would be able to adapt these disclosures to differentopportunities and design criteria.

The above disclosure presented an example of a hierarchy of hardwareelements that each perform the same operation but maintain differentnumbers of bits of precision in their respective outputs. The hierarchyof hardware elements are configured so that outputs of lower-precisionelements provide inputs to higher precision elements. In somesituations, elements within the hierarchy, or some portion thereof, alsomay selectively recycle an output either to itself, or to the input ofanother element. In order to calculate a final output value for asequence of operations, progressively greater precision outputs areachieved by starting from low-precision inputs to the lowest precisionhardware element, and allowing outputs of increasingly higher precisionto propagate through the hierarchy, where propagation also includesmultiple iterations through some subset of the hardware elements. Tyingthis back to the above examples, the hierarchy of the above-examples hastwo levels of multipliers, a lower-precision level and a full-precisionlevel. The full-precision level can recycle outputs to inputs. Iterativerefinement procedures are performed in the two levels by using thelower-precision level once, followed by one or more iterations throughthe full-precision level. However, other implementations may providemore levels of hierarchy, or provide different functions or combinationsthereof within each level.

The above disclosure provided self-consistent examples of delaysexpected through different combinatorial logic elements and arrangementof such logic in a manner selected to cause the different stages ofmultiplier 55 to meet particular timing criteria, and that overall, thecollection of stages meet the example design constraints. These examplesdo not imply that embodiments of the disclosure must be in accordancewith these various details. Indeed, given the same constraints, multipledifferent solutions that meet those constraints can be found. Stillfurther, different implementations may use different fabricationprocesses, different standard cell libraries, or other constraints thatmay cause different relative delays through different elements, andthese differences may call for a different arrangement of pipelinestages or retiming of the pipelines of small multiplier 55 andfull-precision multiplier 25. In these various situations, a person ofordinary skill would be able to adapt these disclosures for thespecifics of those situations. Explicitly, a variety of differentarrangements of combinatorial logic elements to form a carry save addercan satisfy a given set of design constraints and these examples are notlimiting to how a small multiplier 55 according to the disclosure can beconstructed.

Some aspects of the above-example focused on implementing adivide/square root unit, and specifically on producing a reciprocal (1/bfor divide and 1/√b for square root). Although implementations of thedisclosure provide benefits in these contexts, the disclosure also canbe implemented in other contexts. More generally, implementations of thedisclosure provide a reduced precision multiplier circuit that performsone or more multiplications for an operation that requires a series oftwo or more multiplications. Specific benefits result from implementingthis reduced precision multiplier after a LUT operation, in order tocomplete both the LUT action and the reduced precision multiply withinthe same or less time as a full-precision multiply. More generally,aspects disclosed herein can be implemented in circumstances wherepipelined processing occurs on a value generated in an operation thatrequires less time to complete than one cycle through the pipeline, andwhere required precision increases as processing proceeds. Also, whilethe pipeline examples assigned a clock cycle to the LUT that is separatefrom the low-precision first multiply, that is an exampleimplementation, and the operation design point is to provide that thecombined delay of the LUT and the first low-precision multiply completewithin a timeframe required for a full-precision multiply to complete,for a particular implementation.

Modern general purpose processors regularly require in excess of twobillion transistors to be implemented, while graphics processing unitsmay have in excess of five billion transistors. Such transistor countsare likely to increase. Such processors have used these transistors toimplement increasing complex operation reordering, prediction, moreparallelism, larger memories (including more and bigger caches) and soon. As such, it becomes necessary to be able to describe or discusstechnical subject matter concerning such processors, whether generalpurpose or application specific, at a level of detail appropriate to thetechnology being addressed. In general, a hierarchy of concepts isapplied to allow those of ordinary skill to focus on details of thematter being addressed.

For example, high level features, such as what instructions a processorsupports conveys architectural-level detail. When describing high-leveltechnology, such as a programming model, such a level of abstraction isappropriate. Microarchitectural detail describes high level detailconcerning an implementation of an architecture (even as the samemicroarchitecture may be able to execute different ISAs). Yet,microarchitectural detail typically describes different functional unitsand their interrelationship, such as how and when data moves among thesedifferent functional units. As such, referencing these units by theirfunctionality is also an appropriate level of abstraction, rather thanaddressing implementations of these functional units, since each ofthese functional units may themselves comprise hundreds of thousands ormillions of gates. When addressing some particular feature of thesefunctional units, it may be appropriate to identify substituentfunctions of these units, and abstract those, while addressing in moredetail the relevant part of that functional unit.

Functional modules may be composed of circuitry, where such circuitrymay be fixed function, configurable under program control or under otherconfiguration information, or some combination thereof. Functionalmodules themselves thus may be described by the functions that theyperform, to helpfully abstract how some of the constituent portions ofsuch functions may be implemented. In some situations, circuitry andfunctional modules may be described partially in functional terms, andpartially in structural terms. In some situations, the structuralportion of such a description may be described in terms of aconfiguration applied to circuitry or to functional modules, or both.

For example, describing a LookUp Table (LUT), a Carry Save Adder, amultiplexer and so on presents sufficient structural detail to a personof ordinary skill to understand a range of structures that satisfy suchstructural description.

Eventually, a precise logical arrangement of the gates and interconnect(a netlist) implementing these functional units (in the context of theentire processor) can be specified. However, how such logicalarrangement is physically realized in a particular chip (how that logicand interconnect is laid out in a particular design) still may differ indifferent process technology and for a variety of other reasons. Many ofthe details concerning producing netlists for functional units as wellas actual layout are determined using design automation, proceeding froma high level logical description of the logic to be implemented (e.g., a“hardware description language”). Therefore, those of ordinary skill inthe art comprehend that describing functional characteristics of acircuit, group of circuits, or even an entire device also cansufficiently describe structure of such elements.

The term “circuitry” does not imply a single electrically connected setof circuits. Circuitry may be fixed function, configurable, orprogrammable. In general, circuitry implementing a functional unit ismore likely to be configurable, or may be more configurable, thancircuitry implementing a specific portion of a functional unit. Forexample, an Arithmetic Logic Unit (ALU) of a processor may reuse thesame portion of circuitry differently when performing differentarithmetic or logic operations. As such, that portion of circuitry iseffectively circuitry or part of circuitry for each different operation,when configured to perform or otherwise interconnected to perform eachdifferent operation. Such configuration may come from or be based oninstructions, or microcode, for example.

In all these cases, describing portions of a processor in terms of itsfunctionality conveys structure to a person of ordinary skill in theart. In the context of this disclosure, the term “unit” refers, in someimplementations, to a class or group of circuitry that implements thefunctions or functions attributed to that unit. Such circuitry mayimplement additional functions, and so identification of circuitryperforming one function does not mean that the same circuitry, or aportion thereof, cannot also perform other functions. In somecircumstances, the functional unit may be identified, and thenfunctional description of circuitry that performs a certain featuredifferently, or implements a new feature may be described. However, suchstructure also may be produced by a temporary adaptation orconfiguration, such as one caused under program control, microcode, orother source of configuration.

Different approaches to design of circuitry exist, for example,circuitry may be synchronous or asynchronous with respect to a clock.Circuitry may be designed to be static or be dynamic. Different circuitdesign philosophies may be used to implement different functional unitsor parts thereof. Absent some context-specific basis to the contrary,“circuitry” encompasses all such design approaches.

Although circuitry or functional units described herein may be mostfrequently implemented by electrical circuitry, and more particularly,by circuitry that primarily relies on a transistor implemented in asemiconductor as a primary switch element, this term is to be understoodin relation to the technology being disclosed. For example, differentphysical processes may be used in circuitry implementing aspects of thedisclosure, such as optical, nanotubes, micro-electrical mechanicalelements, quantum switches or memory storage, magnetoresistive logicelements, and so on. Although a choice of technology used to constructcircuitry or functional units according to the technology may changeover time, this choice is an implementation decision to be made inaccordance with the then-current state of technology.

Although some subject matter may have been described in languagespecific to examples of structural features and/or method steps, it isto be understood that the subject matter defined in the appended claimsis not necessarily limited to these described features or acts. Forexample, a given structural feature may be subsumed within anotherstructural element, or such feature may be split among or distributed todistinct components. Similarly, an example portion of a process may beachieved as a by-product or concurrently with performance of another actor process, or may be performed as multiple separate acts in someimplementations. As such, implementations according to this disclosureare not limited to those that have a 1:1 correspondence to the examplesdepicted and/or described.

Implementations of the disclosure may be provided for use in embeddedsystems, such as televisions, appliances, vehicles, or personalcomputers, desktop computers, laptop computers, message processors,hand-held devices, multi-processor systems, microprocessor-based orprogrammable consumer electronics, game consoles, network PCs,minicomputers, mainframe computers, mobile telephones, PDAs, tablets andthe like.

In addition to hardware embodiments (e.g., within or coupled to aCentral Processing Unit (“CPU”), microprocessor, microcontroller,digital signal processor, processor core, System on Chip (“SOC”), or anyother programmable or electronic device), implementations may also beembodied in software (e.g., computer readable code, program code,instructions and/or data disposed in any form, such as source, object ormachine language) disposed, for example, in a computer usable (e.g.,readable) medium configured to store the software. Such software canenable, for example, the function, fabrication, modeling, simulation,description, and/or testing of the apparatus and methods describedherein. For example, this can be accomplished through the use of generalprogramming languages (e.g., C, C++), GDSII databases, hardwaredescription languages (HDL) including Verilog HDL, VHDL, SystemCRegister Transfer Level (RTL) and so on, or other available programs,databases, and/or circuit (i.e., schematic) capture tools. Embodimentscan be disposed in computer usable medium including non-transitorymemories such as memories using semiconductor, magnetic disk, opticaldisk, ferrous, resistive memory, and so on.

As specific examples, it is understood that implementations of disclosedapparatuses and methods may be implemented in a semiconductorintellectual property core, such as a microprocessor core, or a portionthereof, embodied in a Hardware Description Language (HDL), that can beused to produce a specific integrated circuit implementation. A computerreadable medium may embody or store such description language data, andthus constitute an article of manufacture. A non-transitory machinereadable medium is an example of computer readable media. Examples ofother embodiments include computer readable media storing RegisterTransfer Language (RTL) description that may be adapted for use in aspecific architecture or microarchitecture implementation. Additionally,the apparatus and methods described herein may be embodied as acombination of hardware and software that configures or programshardware.

Also, in some cases, terminology has been used herein because it isconsidered to more reasonably convey salient points to a person ofordinary skill, but such terminology should not be considered toimpliedly limit a range of implementations encompassed by disclosedexamples and other aspects. A number of examples have been illustratedand described in the preceding disclosure. By necessity, not everyexample can illustrate every aspect, and the examples do not illustrateexclusive compositions of such aspects. Instead, aspects illustrated anddescribed with respect to one figure or example can be used or combinedwith aspects illustrated and described with respect to other figures. Assuch, a person of ordinary skill would understand from these disclosuresthat the above disclosure is not limiting as to constituency ofembodiments according to the claims, and rather the scope of the claimsdefine the breadth and scope of inventive embodiments herein. Thesummary and abstract sections may set forth one or more but not allexemplary embodiments and aspects of the invention within the scope ofthe claims.

What is claimed is:
 1. A method of evaluating an output value of afunction at an input value using an iterative refinement procedurerequiring at least two multiplications for each iteration, comprising:producing an initial approximation of the output value based on theinput value; performing a first multiplication of a first iteration by alimited-precision multiplier; performing, in a full-precisionmultiplier, remaining multiplications for each iteration; and outputtingsaid output value after performing said remaining multiplications. 2.The method of evaluating an output value of a function at an input valueusing an iterative refinement procedure requiring at least twomultiplications for each iteration of claim 1, wherein thefull-precision multiplier requires three clock cycles, the producing ofthe initial approximation of the output value requires one clock cycleand the performing of the first multiplication of the first iterationrequires two clock cycles.
 3. The method of evaluating an output valueof a function at an input value using an iterative refinement procedurerequiring at least two multiplications for each iteration of claim 1,wherein the output value is an approximation of a reciprocal of theinput value and further comprising multiplying the output value with adividend.
 4. The method of evaluating an output value of a function atan input value using an iterative refinement procedure requiring atleast two multiplications for each iteration of claim 1, wherein theoutput value is an approximation of a reciprocal of the square root ofthe input value and further comprising multiplying the output value withthe input value.
 5. The method of evaluating an output value of afunction at an input value using an iterative refinement procedurerequiring at least two multiplications for each iteration of claim 1,further comprising performing two iterations in producing adouble-precision evaluation of the output value.
 6. The method ofevaluating an output value of a function at an input value using aniterative refinement procedure requiring at least two multiplicationsfor each iteration of claim 1, further comprising using thelimited-precision multiplier once and the full-precision multipliertwice in producing a single-precision evaluation of the output value. 7.The method of evaluating an output value of a function at an input valueusing an iterative refinement procedure requiring at least twomultiplications for each iteration of claim 1, in which the initialapproximation has a first number of bits of precision, the first numberof bits of precision being fewer than a required number of bits ofprecision in the output value.
 8. The method of evaluating an outputvalue of a function at an input value using an iterative refinementprocedure requiring at least two multiplications for each iteration ofclaim 1, in which the limited-precision multiplier comprises circuitrycapable of maintaining, in an output, at least twice the number of bitsof precision of the initial approximation and fewer bits of precisionthan required to produce a single-precision floating point mantissa. 9.The method of evaluating an output value of a function at an input valueusing an iterative refinement procedure requiring at least twomultiplications for each iteration of claim 1, in which producing theinitial approximation and performing the first multiplicationcollectively require a first number of clock cycles, and eachmultiplication in the full-precision multiplier requires at least thefirst number of clock cycles.